The cloud hosting environment to run and operate the software can be provided by a variety of methods.
Infrastructure as a service (IAAS). The compute, storage and network services required to run the system. The operating system, patching, application and management are all done by the client.
Platform as a service (PAAS). The low level hardware and operating system are managed by the cloud provider. The customer is responsible for writing the software to use these modules.
Software as a service (SAAS). The customer is reponsible for paying the service and using it..
The service layers can be provided on premise (in the customers office or at a data centre) or via a cloud provider. A few examples are given below.